Chapter 2 One-Dimensional Kinematics

Transcription

1 Review: Chapter 2 One-Dimensional Kinematics Description of motion in one dimension Copyright 2010 Pearson Education, Inc.

2 Review: Motion with Constant Acceleration Free fall: constant acceleration g = 9.81 m/s 2 Copyright 2010 Pearson Education, Inc.

3 Chapter 3 Vectors in Physics Copyright 2010 Pearson Education, Inc.

4 Units of Chapter 3 Scalars Versus Vectors The Components of a Vector Adding and Subtracting Vectors Unit Vectors Position, Displacement, Velocity, and Acceleration Vectors Relative Motion Copyright 2010 Pearson Education, Inc.

5 3-1 Scalars Versus Vectors Scalar: number with units Vector: quantity with magnitude and direction How to get to the library: need to know how far and which way Vector: represented graphically by a line with an arrow Length of the line: magnitude Arrow: direction Copyright 2010 Pearson Education, Inc.

6 3-2 The Components of a Vector Even though you know how far and in which direction the library is, you may not be able to walk there in a straight line: Copyright 2010 Pearson Education, Inc.

7 3-2 The Components of a Vector Can resolve vector into perpendicular components using a two-dimensional coordinate system: Copyright 2010 Pearson Education, Inc.

8 3-2 The Components of a Vector Length, angle, and components can be calculated from each other using trigonometry: A x = A cos θ, A y = A sin θ A: magnitude Or: A = (A x2 + A y2 ) 1/2 θ = tan -1 (A y / A x ) Copyright 2010 Pearson Education, Inc.

9 Copyright 2010 Pearson Education, Inc.

10 Equations used: A x = A cos θ, A y = A sin θ A: magnitude Or: A = (A x2 + A y2 ) 1/2 θ = tan -1 (A y / A x ) Copyright 2010 Pearson Education, Inc. Ask: draw the vectors in blackboard? Express the vectors by unit vectors?

11 3-2 The Components of a Vector Signs of vector components: Copyright 2010 Pearson Education, Inc.

12 Copyright 2010 Pearson Education, Inc. Hint: draw the vectors in x-y plane first!

13 Copyright 2010 Pearson Education, Inc. Ask: 1. Express the vectors by unit vectors? 2. Draw the vectors in blackboard?

14 Copyright 2010 Pearson Education, Inc. Hint: draw the vector in x-y plane first!

15 Copyright 2010 Pearson Education, Inc. Draw the vector in x-y plane in blackboard Express the vectors by unit vectors

16 3-3 Adding and Subtracting Vectors Adding vectors graphically: Place the tail of the second at the head of the first. The sum points from the tail of the first to the head of the last. Copyright 2010 Pearson Education, Inc.

17 Copyright 2010 Pearson Education, Inc.

18 Copyright 2010 Pearson Education, Inc. Adding vectors graphically

19 Question 3.1a Vectors I If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B? a) same magnitude, but can be in any direction b) same magnitude, but must be in the same direction c) different magnitudes, but must be in the same direction d) same magnitude, but must be in opposite directions e) different magnitudes, but must be in opposite directions

20 Question 3.1a Vectors I If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B? a) same magnitude, but can be in any direction b) same magnitude, but must be in the same direction c) different magnitudes, but must be in the same direction d) same magnitude, but must be in opposite directions e) different magnitudes, but must be in opposite directions The magnitudes must be the same, but one vector must be pointing in the opposite direction of the other in order for the sum to come out to zero. You can prove this with the tip-to-tail method.

21 Question 3.1b Vectors II Given that A + B = C, and that lal 2 + lbl 2 = lcl 2, how are vectors A and B oriented with respect to each other? a) they are perpendicular to each other b) they are parallel and in the same direction c) they are parallel but in the opposite direction d) they are at 45 to each other e) they can be at any angle to each other

22 Question 3.1b Vectors II Given that A + B = C, and that lal 2 + lbl 2 = lcl 2, how are vectors A and B oriented with respect to each other? a) they are perpendicular to each other b) they are parallel and in the same direction c) they are parallel but in the opposite direction d) they are at 45 to each other e) they can be at any angle to each other Note that the magnitudes of the vectors satisfy the Pythagorean Theorem. This suggests that they form a right triangle, with vector C as the hypotenuse. Thus, A and B are the legs of the right triangle and are therefore perpendicular.

23 Question 3.1c Vectors III Given that A + B = C, and that lal + lbl = lcl, how are vectors A and B oriented with respect to each other? a) they are perpendicular to each other b) they are parallel and in the same direction c) they are parallel but in the opposite direction d) they are at 45 to each other e) they can be at any angle to each other

24 Question 3.1c Vectors III Given that A + B = C, and that lal + lbl = lcl, how are vectors A and B oriented with respect to each other? a) they are perpendicular to each other b) they are parallel and in the same direction c) they are parallel but in the opposite direction d) they are at 45 to each other e) they can be at any angle to each other The only time vector magnitudes will simply add together is when the direction does not have to be taken into account (i.e., the direction is the same for both vectors). In that case, there is no angle between them to worry about, so vectors A and B must be pointing in the same direction.

25 Adding Vectors Using Components: 1. Find the components of each vector to be added. 2. Add the x- and y-components separately. 3. Find the resultant vector. Copyright 2010 Pearson Education, Inc.

26 3-3 Adding and Subtracting Vectors Subtracting Vectors: The negative of a vector is a vector of the same magnitude pointing in the opposite direction. Here, D= A B. Q: components? Copyright 2010 Pearson Education, Inc.

27 3-4 Unit Vectors Unit vectors are dimensionless vectors of unit length. Copyright 2010 Pearson Education, Inc.

28 Express a vector in terms of unit vectors: A = A x xˆ + A y yˆ Copyright 2010 Pearson Education, Inc.

29 Question 3.2b Vector Components II A certain vector has x and y components that are equal in magnitude. Which of the following is a possible angle for this vector in a standard x-y coordinate system? a) 30 b) 180 c) 90 d) 60 e) 45

30 Question 3.2b Vector Components II A certain vector has x and y components that are equal in magnitude. Which of the following is a possible angle for this vector in a standard x-y coordinate system? a) 30 b) 180 c) 90 d) 60 e) 45 The angle of the vector is given by tan θ = y/x. Thus, tan θ = 1 in this case if x and y are equal, which means that the angle must be 45.

31 Copyright 2010 Pearson Education, Inc. Adding and Subtracting Vectors Formula using unit vectors y A x A A y x ˆ ˆ + = y B x B B y x ˆ ˆ + = y B A x B A B A y y x x ˆ ) ( ) ˆ ( = +

32 3-4 Unit Vectors Multiplying vectors by scalars: the multiplier changes the length, and the sign indicates the direction. α A = ( αa ) xˆ + ( αa Copyright 2010 Pearson Education, Inc. x y ) yˆ

33 Question 3.2a Vector Components I If each component of a vector is doubled, what happens to the angle of that vector? a) it doubles b) it increases, but by less than double c) it does not change d) it is reduced by half e) it decreases, but not as much as half

34 Question 3.2a Vector Components I If each component of a vector is doubled, what happens to the angle of that vector? a) it doubles b) it increases, but by less than double c) it does not change d) it is reduced by half e) it decreases, but not as much as half The magnitude of the vector clearly doubles if each of its components is doubled. But the angle of the vector is given by tan θ = 2y/2x, which is the same as tan θ = y/x (the original angle). Follow-up up: if you double one component and not the other, how would the angle change?

35 Summary of Chapter 3a Scalar: number, with appropriate units Vector: quantity with magnitude and direction Vector components: A x = A cos θ, B y = B sin θ Magnitude: A = (A x2 + A y2 ) 1/2 Direction: θ = tan -1 (A y / A x ) Graphical vector addition: Place tail of second at head of first; sum points from tail of first to head of last

36 Summary of Chapter 3 Component method: add components of individual vectors, then find magnitude and direction Unit vectors are dimensionless and of unit length

37 Copyright 2010 Pearson Education, Inc.